scholarly journals Heavy handed quest for fixed points in multiple coupling scalar theories in the ε expansion

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Hugh Osborn ◽  
Andreas Stergiou
Keyword(s):  
Fixed Points ◽  
Index Case ◽  
Symmetric Tensors ◽  
Number Of Solutions ◽  
Stability Matrix ◽  
Negative Eigenvalues ◽  
Large Numbers ◽  
Quadratic Invariants ◽  
Scalar Theories ◽  
The Stability

Abstract The tensorial equations for non trivial fully interacting fixed points at lowest order in the ε expansion in 4 − ε and 3 − ε dimensions are analysed for N-component fields and corresponding multi-index couplings λ which are symmetric tensors with four or six indices. Both analytic and numerical methods are used. For N = 5, 6, 7 in the four-index case large numbers of irrational fixed points are found numerically where ‖λ‖2 is close to the bound found by Rychkov and Stergiou [1]. No solutions, other than those already known, are found which saturate the bound. These examples in general do not have unique quadratic invariants in the fields. For N ⩾ 6 the stability matrix in the full space of couplings always has negative eigenvalues. In the six index case the numerical search generates a very large number of solutions for N = 5.

scholarly journals On polytopes and generalizations of the KLT relations

2020 ◽  
Vol 2020 (12) ◽  
Author(s):  
Nikhil Kalyanapuram
Keyword(s):  
Scattering Amplitudes ◽  
Large Class ◽  
Moduli Space ◽  
Differential Forms ◽  
Natural Generalization ◽  
Intersection Theory ◽  
Hyperplane Arrangements ◽  
Number Of Solutions ◽  
Scalar Theories ◽  
Double Copy

Abstract We combine the technology of the theory of polytopes and twisted intersection theory to derive a large class of double copy relations that generalize the classical relations due to Kawai, Lewellen and Tye (KLT). To do this, we first study a generalization of the scattering equations of Cachazo, He and Yuan. While the scattering equations were defined on ℳ0, n — the moduli space of marked Riemann spheres — the new scattering equations are defined on polytopes known as accordiohedra, realized as hyperplane arrangements. These polytopes encode as patterns of intersection the scattering amplitudes of generic scalar theories. The twisted period relations of such intersection numbers provide a vast generalization of the KLT relations. Differential forms dual to the bounded chambers of the hyperplane arrangements furnish a natural generalization of the Bern-Carrasco-Johansson (BCJ) basis, the number of which can be determined by counting the number of solutions of the generalized scattering equations. In this work the focus is on a generalization of the BCJ expansion to generic scalar theories, although we use the labels KLT and BCJ interchangeably.

scholarly journals Emergent chiral symmetry in a three-dimensional interacting Dirac liquid

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
András L. Szabó ◽  
Bitan Roy
Keyword(s):  
Fixed Points ◽  
Chiral Symmetry ◽  
Rotational Symmetry ◽  
Three Dimensional ◽  
Spatial Dimension ◽  
Dirac Fermions ◽  
Electronic Interactions ◽  
Emergent Phenomena ◽  
Ordered Phases ◽  
The Stability

Abstract We compute the effects of strong Hubbardlike local electronic interactions on three-dimensional four-component massless Dirac fermions, which in a noninteracting system possess a microscopic global U(1) ⊗ SU(2) chiral symmetry. A concrete lattice realization of such chiral Dirac excitations is presented, and the role of electron-electron interactions is studied by performing a field theoretic renormalization group (RG) analysis, controlled by a small parameter ϵ with ϵ = d−1, about the lower-critical one spatial dimension. Besides the noninteracting Gaussian fixed point, the system supports four quantum critical and four bicritical points at nonvanishing interaction couplings ∼ ϵ. Even though the chiral symmetry is absent in the interacting model, it gets restored (either partially or fully) at various RG fixed points as emergent phenomena. A representative cut of the global phase diagram displays a confluence of scalar and pseudoscalar excitonic and superconducting (such as the s-wave and p-wave) mass ordered phases, manifesting restoration of (a) chiral U(1) symmetry between two excitonic masses for repulsive interactions and (b) pseudospin SU(2) symmetry between scalar or pseudoscalar excitonic and superconducting masses for attractive interactions. Finally, we perturbatively study the effects of weak rotational symmetry breaking on the stability of various RG fixed points.

scholarly journals Invariant curves around a parabolic fixed point at infinity

1990 ◽  
Vol 10 (2) ◽  
pp. 209-229 ◽  
Author(s):  
Dov Aharonov ◽  
Uri Elias
Keyword(s):  
Fixed Point ◽  
Fixed Points ◽  
Invariant Curves ◽  
The Stability ◽  
Area Preserving ◽  
Parabolic Fixed Point

AbstractThe stability of a fixed point of an area-preserving transformation in the plane is characterized by the invariant curves which surround it. The existence of invariant curves had been extensively studied for elliptic fixed points. Here we study the similar problem for parabolic fixed points. In particular we are interested in the case where the fixed point is at infinity.

scholarly journals Dynamics of f(R) gravity models and asymmetry of time

2018 ◽  
Vol 27 (02) ◽  
pp. 1850002 ◽  
Author(s):  
Murli Manohar Verma ◽  
Bal Krishna Yadav
Keyword(s):  
Fixed Points ◽  
Quadratic Forms ◽  
Scale Factor ◽  
Gravity Models ◽  
Field Equations ◽  
Classical Level ◽  
The Matrix ◽  
The Stability ◽  
Effects Of Radiation ◽  
Linear And Quadratic Forms

We solve the field equations of modified gravity for [Formula: see text] model in metric formalism. Further, we obtain the fixed points of the dynamical system in phase-space analysis of [Formula: see text] models, both with and without the effects of radiation. The stability of these points is studied against the perturbations in a smooth spatial background by applying the conditions on the eigenvalues of the matrix obtained in the linearized first-order differential equations. Following this, these fixed points are used for analyzing the dynamics of the system during the radiation, matter and acceleration-dominated phases of the universe. Certain linear and quadratic forms of [Formula: see text] are determined from the geometrical and physical considerations and the behavior of the scale factor is found for those forms. Further, we also determine the Hubble parameter [Formula: see text], the Ricci scalar [Formula: see text] and the scale factor [Formula: see text] for these cosmic phases. We show the emergence of an asymmetry of time from the dynamics of the scalar field exclusively owing to the [Formula: see text] gravity in the Einstein frame that may lead to an arrow of time at a classical level.

New fixed point of on-orbit calibration scale based on the In-Bi eutectic alloy for application in novel high-stable space-borne standard sources

2021 ◽  
pp. 32-37
Author(s):  
Andrei A. Burdakin ◽  
Valerii R. Gavrilov ◽  
Ekaterina A. Us ◽  
Vitalii S. Bormashov
Keyword(s):  
Phase Transition ◽  
Fixed Point ◽  
Fixed Points ◽  
Eutectic Alloy ◽  
Earth Observation ◽  
Melt Temperature ◽  
Phase Transition Temperatures ◽  
Set Up ◽  
The Stability ◽  
Observation Space

The problem of ensuring stability of Earth observation space-borne instruments undertaking long-term temperature measurements within thermal IR spectral range is described. For in-flight reliable control of the space-borne IR instruments characteristics the stability of onboard reference sources should be improved. The function of these high-stable sources will be executed by novel onboard blackbodies, incorporating the melt↔freeze phase transition phenomenon, currently being developed. As a part of these works the task of realizing an on-orbit calibration scale within the dynamic temperature range of Earth observation systems 210−350 K based on fixed-point phase transition temperatures of a number of potentially suitable substances is advanced. The corresponding series of the onboard reference blackbodies will be set up on the basis of the on-orbit calibration scale fixed points. It is shown that the achievement of the target lies in carrying out a number of in-flight experiments with the selected fixed points and the prospective onboard fixed-point blackbodies prototypes. The new In-Bi eutectic alloy melt temperature fixed point (~345 K) is proposed as the significant fixed points of the future on-orbit calibration scale. The results of the new fixed point preliminary laboratory studies have been analyzed. The results allowed to start preparation of the in-flight experiments investigating the In-Bi alloy for the purpose of its application in the novel onboard reference sources.

scholarly journals Numerical Reckoning Fixed Points of (ρE)-Type Mappings in Modular Vector Spaces

Mathematics ◽  
2019 ◽  
Vol 7 (5) ◽  
pp. 390 ◽  
Author(s):  
Wissam Kassab ◽  
Teodor Ţurcanu
Keyword(s):  
Fixed Points ◽  
Banach Spaces ◽  
Iteration Process ◽  
Existence Results ◽  
Data Dependence ◽  
Vector Spaces ◽  
The Stability ◽  
Recent Version

In this paper, we study an iteration process introduced by Thakur et al. for Suzuki mappings in Banach spaces, in the new context of modular vector spaces. We establish existence results for a more recent version of Suzuki generalized non-expansive mappings. The stability and data dependence of the scheme for ρ -contractions is studied as well.

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